The mean of a normal probability distribution is 400 pounds. The standard deviat

Question

The mean of a normal probability distribution is 400 pounds. The standard deviation is 10 pounds.
a) what is the area between 415 pounds and the mean of 400 pounds?
b) what is the area between the mean and 395 pounds?
c) what is the probability of selecting a value at a random and discovering that it has a value of less than 395 pounds?

This is my first course in statistics and I'm pretty much lost.. Please help me! I've got 6 similar questions and if I get this one I'll be able to work with the others. Thank you!

Explanation / Answer

I would love to help. Any problems in Normal Probability Distribution normally require the use of a Normal Probability Distribution Table (most of the time this is located at the back of your book). Most tables are cumulative, meaning they add the probabilities up as they go. This means, if you look up a value (say 1.34) then you're going to be given the probability of being at value or less. So let's make sure we rephrase each question in terms of P(X < x) (you should pick up the pattern as we go). The best part of these problems is that they require no calculus (yay!). For extra help, consider drawing the normal curve for each problem.

From the problem we have X~N(400,10).

a) P(X is between 415 and 400) Think about the cumulative curve, if we add these, we'll be double counting the values up to 400. So to get the values in between, we should subtract.

= P(X < 415) - P(X < 400)

To use the standard normal table, we need to standardize our distribution, which means we just shift it so that the standard values in the table can apply. Since the random variable X no longer applies, for this standardized version we use the random variable Z.

= P(Z < (415-)/) - 0.5 = P(Z < (415-400)/10)

Remember that at the average, 0.5 of the values are above and 0.5 of the values are below.

= P(Z < 1.5) - 0.5 = 0.9332 - 0.5 = 0.4332

Now that we've gone through these step by step, we can solve the rest of the problems in a similar way:

b) P(X is between 395 and 400) = P(X < 400) - P(X < 395) = 0.5 - P(Z < (395-)/) = 0.5 - P(Z < -0.5)

Think of the symmetry of the normal curve. The probability of being at or below -0.5 is going to be the same as the probability of being above +0.5. Well the probability of being above +0.5 is going to be the opposite of being at or below 0.5. SO:

0.5 - P(Z<-0.5) = 0.5 - (1 - P(Z < 0.5)) = 0.5 - (1 - 0.6519) = 0.5 - 0.3481 = 0.1519

c)P(A randomly selected value, or, X, is less than 395) = P( X < 395) = P(Z < (395-400)/10) = P(Z < -0.5)

We solved this earlier: = 1 - P(Z < 0.5) = 0.3481

If you have any further questions, let me know!

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